Definition:Sigma-Algebra Generated by Collection of Subsets

Definition

Let $X$ be a set.

Let $\mathcal G \subseteq \powerset X$ be a collection of subsets of $X$.

Definition 1

The $\sigma$-algebra generated by $\mathcal G$, $\map \sigma {\mathcal G}$, is the smallest $\sigma$-algebra on $X$ that contains $\mathcal G$.

That is, $\map \sigma {\mathcal G}$ is subject to:

$(1): \quad \mathcal G \subseteq \map \sigma {\mathcal G}$
$(2): \quad \mathcal G \subseteq \Sigma \implies \map \sigma {\mathcal G} \subseteq \Sigma$ for any $\sigma$-algebra $\Sigma$ on $X$

Definition 2

Then the $\sigma$-algebra generated by $\mathcal G$, $\map \sigma {\mathcal G}$, is the intersection of all $\sigma$-algebras on $X$ that contain $\mathcal G$.

Generator

One says that $\mathcal G$ is a generator for $\map \sigma {\mathcal G}$.

Also, elements $G$ of $\mathcal G$ may be called generators.

Also denoted as

Variations of the letter "$M$" can be seen for the $\sigma$-algebra generated by $\mathcal G$:

$\map {\mathcal M} {\mathcal G}$
$\map {\mathscr M} {\mathcal G}$